Let $X$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. A subgroup $H$ of a finite group $G$ is called a submaximal $X$-subgroup if there exists an isomorphic embedding $\phi : G \hookrightarrow G^*$ of $G$ into some finite group $G^*$ under which $G^{\phi}$ is subnormal in $G^*$ and $H^{\phi} = K \cap G^{\phi}$ for some maximal $X$-subgroup $K$ of $G^*$. In the case where $X$ coincides with the class of all $\pi $-groups for some set $\pi $ of prime numbers, submaximal $X$-subgroups are called submaximal $\pi $-subgroups. Here, the authors prove properties of maximal and submaximal $X$- and $\pi $-subgroups and discuss some open questions. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal $X$-subgroups are conjugate in a finite group G in which all maximal $X$-subgroups are conjugate?